Exponential approximation stationary states and local stability

Exponential approximation stationary states and local stability 

Before proceeding, let us quickly summarize the operations carried out so far in this chapter. We have taken a chemical model consisting of two first-order reactions. The first step, which produces the intermediate A, is essentially a slow process and so we have made the pool chemical approximation for the concentration of the original reactant P. The second step converts A to a final product, releasing heat. The non-linear temperature dependence of the rate of this second step has been approximated by a simple exponential function of the consequent dimensionless temperature rise. 

The mass- and energy-balance equations can now be written in the explicit form 

Again we may consider that the dimensionless concentration of the reactant fx will automatically vary during the course of a given experiment with the consumption of P, whilst k might vary between successive experiments. 

We will regard a and 6 as functions of the reactant concentration as expressed in fi and assume that they change on a fast timescale compared with reactant consumption (small e), i.e. we apply the pseudo-stationary-state hypothesis. The pseudo-stationary-state condition da/dt = dO/dz = 0 yields the following simultaneous equations  

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